High School Algebra I Study Guide for the STAAR test

Exponential Functions and Equations

Exponential functions and equations are the last class of functions and equations you will need to know. Six of the 54 questions on the test assess these skills.

Domain and Range

An exponential function is in the form \(y=ab^x\), where \(b\gt0, \;b \neq 1\), and \(a\) is a real number. In general, its graph increases at an increasing rate or decreases at an increasing rate, as shown below.

For the left graph: \(f(x)=ab^x,\;a \gt 0,\;b \gt 1\)
For the right graph: \(f(x) = ab^x, \;a \gt 0,\; 0 \lt b \lt 1\)

There is a horizontal asymptote at \(x=0\).
If \(a\) is negative, the graph is reflected across the \(x\)-axis.
The domain is all Real numbers, and the range is \(y \ge 0\) if \(a \gt 0\) or \(y\le0\) if \(a \lt 0\).

For example:

State the domain and range for \(f(x)=-3 \cdot 2^x\).

Since \(a\) is negative, the graph is reflected across the \(x\)-axis, making the range less than or equal to zero, so the domain is \((-\infty, \infty)\) and the range is \((-\infty, 0)\) in interval notation.

Value Meanings

The value of \(b\) is similar to the slope of a line in that its value tells you the exponential rate at which the function is increasing or decreasing.

For example:

Interpret the meaning of \(a\) and \(b\) in this situation:

Suppose the function \(f(x)=100(1.1)^x\) is used to represent an amount of money, in dollars. You have an investment in a mutual fund you made where \(x\) is time, in years.

\(a\) is the initial amount of the investment. Remember, “initially” means \(x=0\), so \(f(0)=100(1.1)^0=100(1)=100\). The term \(b\) is the rate at which the investment is growing each year. The investment is increasing, since \(b \gt 0\).

Writing Exponential Functions

In growth and decay situations, look for words such as initial or original to find the value for \(a\), and words such as increasing, decreasing, or rate to find the value for \(b\) when writing an exponential function for the given situation.

Here’s an example:

The amount of water in a reservoir is decreasing at \(1.5 \%\) each year. The reservoir initially has \(200\) billion gallons of water in it. Write an exponential function, \(w (t)\) where \(w\) is the amount of water in gallons, and \(t\) is time in years.

Since the amount of water is decreasing at \(1.5\%\), we must subtract \(1.5\%\) from \(100\%\) or \(1.00 -0.015 = 0.985\) to find \(b\). The initial amount of \(200\) is \(a\). Then \(w(t)=200(0.985)^t\).

Graph Attributes

You may also need to know how to graph an exponential function for a given situation, identifying key attributes such as the intercepts, and asymptotes of the graph.

Suppose the growth of an exponential function, \(f\) is doubling for each domain value. Sketch a graph of the function \(f(0) = 3\).

Since the \(f\) is doubling in value, its graph is increasing at an increasing rate. Also, since \(f(0) = 3\), the \(y\)-intercept of the graph. Graphing a few select points such as:

\[f(-1)=3(2)^{-1}= \frac{2}{3}\]

and

\[f(1)=3(2)^1=6\]

and

\[f(2)=3(2)^2=3 \cdot 4=12\]

we can make a sketch like the one below.

\[f(x)=3(2)^x\]

Best Fit

Lastly, technology, such as the graphing calculator, will be needed to provide an exponential function of best fit in a given situation.

The table below shows the amount of water, \(y\), in gallons that are left as the water evaporates from a boiling pot after \(t\) minutes. Write an exponential regression that models the amount of water left in the pot after \(t\) minutes.

\[\begin{array}{|c|c|c|} \hline t \;\text{(min.)} & y\; \text{(gallons)} \\ \hline \text{0} & \text{10} \\ \hline \text{2} & \text{9.2} \\ \hline \text{7} & \text{7.2} \\ \hline \text{10} & \text{5.8} \\ \hline \end{array}\]

Input the time in L1 and the number of gallons left in L2 by entering STAT and then EDIT on your graphing calculator. Since we are told the data fits an exponential function, enter STAT then over to CALC and enter 0:ExpReg. A summary of what you did will appear on the display. Scroll down to Calculate, and the exponential function will appear.

The answer is \(y(t)=10(.95)^t\).